Suppose $R$ is a ring and $R[x]$ is the ring of polynomials in the indeterminate $x$ with coefficients from $R$. The characteristic of a ring is the smallest positive integer $n$ such that $n \cdot r =0$ for all $r$ in $R$, or $0$ if no such $n$ exists.
I'm interested in the truth of statement "the characteristic of $R$ is equal to the characteristic of $R[x]$."
If $R$ has unity then I believe the statement is true. Likewise, if $R$ has characteristic $0$ I believe the statement is true. The last case is then rings with no unity and characteristic $n>0$. However, no examples of this kind come to mind.
My questions are then:
Is the statement true?
If so, is there an easier way to show it besides a case breakdown?
Are there rings with no unity and positive characteristic?